Explanation:
To find the value of the expression F(1) F(2) F(3) F(4) ... F(1000), we need to determine the pattern or relationship between the terms.

Identifying the Pattern:
Let's first calculate the values of F(1), F(2), F(3), and F(4) to observe any pattern:

F(1) = 1
F(2) = 2
F(3) = 3
F(4) = 4

From these values, we can see that F(n) = n for all positive integers n.

Using the Pattern:
Since F(n) = n for all positive integers n, we can rewrite the expression as:

F(1) F(2) F(3) F(4) ... F(1000) = 1 * 2 * 3 * 4 * ... * 1000

This is the product of all positive integers from 1 to 1000, which can be written as 1000!.

Calculating the Value:
To calculate the value of 1000!, we can use the concept of factorial. The factorial of a positive integer n is the product of all positive integers from 1 to n.

1000! = 1 * 2 * 3 * 4 * ... * 1000

Since the factorial of a number grows very quickly, it is not practical to calculate it manually. However, we can use a calculator or programming language to find the value.

Using a calculator or programming language, we can find that the value of 1000! is a very large number. However, we don't need to find the exact value to answer the question.

Determining the Last Digit:
To determine the last digit of the value, we can observe that multiplying any number by 1 does not change its last digit. Therefore, when we multiply all positive integers from 1 to 1000, the last digit of the resulting value will be the same as the last digit of 1.

Hence, the last digit of 1000! is 1.

Choosing the Correct Answer:
Now, let's look at the options given:

a) 2001
b) 1999
c) 2004
d) 1997

Since we have determined that the last digit of 1000! is 1, the only option that ends with 1 is option a) 2001.

Therefore, the correct answer is option a) 2001.